Analytical solution of the geodesic equation in Kerr-(anti-) de Sitter space-times

The complete analytical solutions of the geodesic equations in Kerr-de Sitter and Kerr-anti-de Sitter space-times are presented. They are expressed in terms of Weierstrass elliptic $\wp$, $\zeta$, and $\sigma$ functions as well as hyperelliptic Kleinian $\sigma$ functions restricted to the one-dimensional $\theta$ divisor. We analyze the dependency of timelike geodesics on the parameters of the space-time metric and the test-particle and compare the results with the situation in Kerr space-time with vanishing cosmological constant. Furthermore, we systematically can find all last stable spherical and circular orbits and derive the expressions of the deflection angle of flyby orbits, the orbital frequencies of bound orbits, the periastron shift, and the Lense-Thirring effect.

Figure 1

Regions of slow and fast Kerr-de Sitter (KdS) for different values of $\Lambda$. Left: the black curve corresponds to $\Lambda =0$, the gray curve to $\Lambda =-{10}^{-5}$. Below the line we have slow KdS and fast above. Right: $\Lambda ={10}^{-5}$. The region bounded by the two curves corresponds to slow KdS, outside to fast.

Figure 2

Typical regions of different types of $\theta$-motion. Here, $\overline{M}=2$, $\overline{\Lambda }={10}^{-5}$, and $\overline{K}=3$. The transition from slow Kerr-de Sitter (left, $\overline{a}=0.4$) to fast Kerr-de Sitter (right, $\overline{a}=0.8$) is continuous.

Figure 3

Typical examples of $\Theta (\theta )$ in the two regions (b) [left] and (d) [right]. Allowed values of $\theta$ are given by $\Theta (\theta )\ge 0$.

Figure 4

Typical examples of $R(\overline{r})$ for $\Lambda >0$ and for different numbers of real zeros. Allowed values of $\overline{r}$ are given by $R(\overline{r})\ge 0$.

Figure 5

Regions of different types of $\overline{r}$ motion. Here, $\overline{a}=0.4$ and $\overline{K}=3$. Left column $\overline{\Lambda }={10}^{-5}$, middle $\overline{\Lambda }=0$ and right column $\overline{\Lambda }=-{10}^{-5}$. Note that in the upper row for $\overline{\Lambda }>0$ the vertical line has a maximum and for $\overline{\Lambda }<0$ a minimum. For $\overline{\Lambda }=0$ the vertical line is not related to a change of the number of real zeros but a switch from $\underset{\overline{r}\to \infty }{lim}R(\overline{r})=\infty$ to $\underset{\overline{r}\to \infty }{lim}R(\overline{r})=-\infty$. Boundaries of the regions correspond to multiple zeros in $R(\overline{r})$ and, therefore, to the occurrence of a stable or unstable spherical orbit. A detailed view including the results from the $\theta$ motion can be seen in the lower row (note the rescaled axes).

Figure 6

Highlighted view of region (d) of $\theta$ motion combined with $r$ motion for $\overline{a}=0.4$, $\overline{\Lambda }={10}^{-5}$, and $\overline{K}=3$. Regions (Id) and (IId) correspond to $Q<0$ and, thus, to crossover orbits, whereas region (IIb) corresponds to $Q>0$. Note the differently scaled axes compared to Fig. 5.

Figure 7

Deformation of regions of $r$ motion for $\overline{a}=0.4$, $\overline{\Lambda }={10}^{-5}$, and different $\overline{K}$. From left to right $\overline{K}=0$, $\overline{K}=1$, $\overline{K}=5$, and $\overline{K}=9$. For $\overline{K}=0$ region (d) of $\theta$ motion vanishes and region (b) reduces to the line ${\overline{L}}_z=\overline{a}E$, which, therefore, is the only allowed parameter region and corresponds to $Q=0$ and $\theta \equiv \frac{\pi }{2}$, see Thm. 2. Around this line ${\overline{L}}_z=\overline{a}E$ additional parts of regions (III) and (IV) appear not present for $\overline{K}>0$.

Figure 8

Flyby orbits for $\overline{\Lambda }={10}^{-5}$, $\overline{a}=0.4$, and $\overline{K}=3$. For $\Lambda =0$ the parameters of (c) would have been located in region (Vb) and the parameters of (d) in region (IIIb). Light grey cones correspond to extremal $\theta$ and dark grey spheres to horizons.

Figure 9

Precession of the orbital plane for a bound orbit with $\overline{\Lambda }={10}^{-5}$, $\overline{a}=0.4$, $\overline{K}=4.5$, ${\overline{L}}_z=-0.5$, and $E^2=0.96$ [region (IIIb)]. As in Fig. 8, the cones and spheres correspond to extremal $\theta$ and horizons.

Figure 10

(a): Last Stable Spherical Orbit at $\overline{r}(\gamma )\equiv 2.98982782315$. The corresponding parameter values are (approximately) $E^2=0.88830740016$ and ${\overline{L}}_z=0.00154804028$ [lower left corner on boundary of region (IIIb)]. (b): Stable spherical orbit at $\overline{r}(\gamma )\equiv 0.14$ within the inner Cauchy horizon. Here, $E^2=124.76756414675$ and ${\overline{L}}_z=5.879796033955$ [region (IIb)].

Figure 11

Unstable spherical orbit with $\overline{r}(\gamma )\equiv 1.5$ (left) and asymptotic approach (right). Here, $\overline{a}=0.3$, $\overline{\Lambda }={10}^{-5}$, $\overline{K}=2$, $E=0.9720311146$ and ${\overline{L}}_z=2.5677957370$ [boundary of region (IIIb)].